Properties

Degree 1
Conductor 229
Sign $0.998 - 0.0581i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.324 − 0.945i)2-s + (−0.986 + 0.164i)3-s + (−0.789 − 0.614i)4-s + (0.879 + 0.475i)5-s + (−0.164 + 0.986i)6-s + (−0.915 − 0.401i)7-s + (−0.837 + 0.546i)8-s + (0.945 − 0.324i)9-s + (0.735 − 0.677i)10-s + (0.0825 − 0.996i)11-s + (0.879 + 0.475i)12-s + (−0.475 + 0.879i)13-s + (−0.677 + 0.735i)14-s + (−0.945 − 0.324i)15-s + (0.245 + 0.969i)16-s + (−0.879 − 0.475i)17-s + ⋯
L(s,χ)  = 1  + (0.324 − 0.945i)2-s + (−0.986 + 0.164i)3-s + (−0.789 − 0.614i)4-s + (0.879 + 0.475i)5-s + (−0.164 + 0.986i)6-s + (−0.915 − 0.401i)7-s + (−0.837 + 0.546i)8-s + (0.945 − 0.324i)9-s + (0.735 − 0.677i)10-s + (0.0825 − 0.996i)11-s + (0.879 + 0.475i)12-s + (−0.475 + 0.879i)13-s + (−0.677 + 0.735i)14-s + (−0.945 − 0.324i)15-s + (0.245 + 0.969i)16-s + (−0.879 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.998 - 0.0581i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.998 - 0.0581i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $0.998 - 0.0581i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (8, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ 0.998 - 0.0581i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.086424542 - 0.03162034083i$
$L(\frac12,\chi)$  $\approx$  $1.086424542 - 0.03162034083i$
$L(\chi,1)$  $\approx$  0.7959878735 - 0.2827342660i
$L(1,\chi)$  $\approx$  0.7959878735 - 0.2827342660i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.795773048405265408113305427141, −25.09638950916356930520361308312, −24.412997753437167435544265647098, −23.31836303714018821745013953477, −22.47478370852127858259424865021, −21.97930091943721006051059818241, −20.91827117421372217013284865920, −19.3900948888659871561471317302, −18.12356218832977994310345935562, −17.39312988514078551099358643642, −16.89604708138910576569621188283, −15.71053934406173973927315993393, −15.04318671864093059323659251287, −13.41815435854838911137075630223, −12.80953027223875070911102879119, −12.20493343559026342404095095312, −10.363679264146155542877408425615, −9.55237441242995642263082538623, −8.36169871791076846962337659933, −6.767727149058991568533493557144, −6.35471011435505049845183952, −5.19625287735897921952347552761, −4.447778584971324925676036760590, −2.47089928087530322723042926960, −0.46461730498994548261281192738, 0.9506148338464927493213372287, 2.46067017965072627140329543605, 3.74870808243042332591350980424, 4.94306193014106329760005369330, 6.1264074532866542105079097927, 6.75311231804302075162386718117, 9.04942051033023440756031633126, 9.86678413729683593903348978413, 10.7181335496201030635653273521, 11.46152243315879175407600562058, 12.66117437288635891661047121240, 13.47172641814503774018819812460, 14.33518702179313899004828499396, 15.779717793877570647424596367261, 16.92669679784899695065855944300, 17.67528216758089601013391585413, 18.83888013737421068161072885715, 19.35833675663603858283176884779, 20.88768062986052764787496723842, 21.687659032953283115450307463703, 22.17633690573720603571679802743, 23.09739612023082452254677936362, 23.87350437783755932425811689160, 25.119584170409691381266574558056, 26.6918260754485552310625872624

Graph of the $Z$-function along the critical line